Question: All of the 5th grade teachers and students from Covington went on a field trip to an art museum. Tickets were $$8.00$ each for teachers and $$3.50$ each for students, and the group paid $$30.00$ in total. The next month, the same group visited a natural history museum where the tickets cost $$16.00$ each for teachers and $$12.50$ each for students, and the group paid $$82.00$ in total. Find the number of teachers and students on the field trips.
Answer: Let $x$ equal the number of teachers and $y$ equal the number of students. The system of equations is: ${8x+3.5y = 30}$ ${16x+12.5y = 82}$ Solve for $x$ and $y$ using elimination. Multiply the top equation by $-2$ ${-16x-7y = -60}$ ${16x+12.5y = 82}$ Add the top and bottom equations together. $ 5.5y = 22 $ $ y = \dfrac{22}{5.5}$ ${y = 4}$ Now that you know ${y = 4}$ , plug it back into $ {8x+3.5y = 30}$ to find $x$ ${8x + 3.5}{(4)}{= 30}$ $8x+14 = 30$ $8x = 16$ $x = \dfrac{16}{8}$ ${x = 2}$ You can also plug ${y = 4}$ into $ {16x+12.5y = 82}$ and get the same answer for $x$ ${16x + 12.5}{(4)}{= 82}$ ${x = 2}$ There were $2$ teachers and $4$ students on the field trips.